Imaginary difference to sum

[AntonioLao]

The initial crowning achievement of a branch of mathematics known as complex analysis is the discovery of contour integration by Cauchy in 1814. With his theory of complex integration many of those impossible definite integrals of real analysis can now be easily determined.

Augustin-Louis Cauchy (1789-1857) started out as a civil engineer and by 1810 was helping Napoleon building a naval base for an assault against England. However, a year later because of poor health caused by childhood malnutrition he abandoned military engineering to begin a lifelong career as a mathematician. He passed on a legacy of about 800 papers and 7 books on mathematics. These magnum opuses second only to Euler’s. As an adult he appeared lacking in common sense and projected himself with naïve, childish, even rude clumsy behavior. The Norwegian mathematician Niels Abel visited him in 1826 with the 1st impression of him as a religious bigot and much worse. Today, nobody cares about Cauchy as a man but every consummate student of mathematics definitely agrees that he was a mathematical genius.

Complex integration is only possible for any complex function that is analytic. For a function to be analytic its derivative must exists. However, in the 2-dimensional complex plane there are infinitely many ways for a complex variable z to vanish: Dz®0, its infinitesimal difference approaches zero. Moreover, the derivative needs the most condition-free definition and insists it matters not how it approaches zero. But of course being free of conditions is a philosophical premonition. For mathematics it is free to define anything anyone wishes. The important thing is for the definition to solve difficult problems even if there are restrictions. These restrictive conditions are known as the Cauchy-Riemann partial differential equations: ¶u/¶x=¶v/¶y and ¶u/¶y=-¶v/¶x where u is the real part and v is the imaginary part of a complex function: f(z)=u+iv and z=x+iy such that the integral of f(z) around any nonintersecting closed path is zero:òf(z)dz=0 known as Cauchy’s first continuous sum theorem.

[mkirkpatrick]

To find the sum of all we need to imagine all!




regards michael.

[AntonioLao]

As Cauchy knew in 1814, they more you imagine the closer you get to zero. He called it his first integral theorem of complex imaginary analysis